So last week we had the ‘Truth be Told’ conference in Amsterdam, bringing together top-notch researchers on truth coming both from formal and from philosophical quarters (F. A. Müller wrote a short report of the conference here). Indeed, the main purpose of the conference was really to get these different groups of people to talk to each other, in particular to discuss the potential philosophical significance of the many interesting technical results on axiomatic theories of truth.
I must admit that, as of lately, I had become a bit skeptical concerning the philosophical significance of many of the developments in the field of axiomatic theories of truth. My feeling was that it had become an independent field of mathematical inquiry, as it were, scarcely related to its original target-phenomenon, namely the concept of truth. With Tarski, the connection between the formal and the conceptual level was still very palpable (Tarski was always adamant to formulate clear conditions of material adequacy for his formal theories, which were drawn directly from the conceptual level), but at least since Kripke’s fixed-point constructions, things had become increasingly baroque.
True enough, one well-known result with apparent philosophical significance is the non-conservativeness of some theories of PA (Peano Arithmetic) when a truth predicate is added to them (PA*), with respect to the same theory without the predicate (i.e., there are statements that you can prove in PA* which you cannot prove in PA). The technical details are a bit complicated (essentially, the truth predicate behaves like a second-order quantifier), but as Horsten and others have argued, this seems to suggest that at least some forms of deflationism may be wrongheaded, given that the truth predicate does seem to add something of ‘substance’ to the system in question.
In this spirit, one of the most interesting talks at the conference, in my opinion, was the one by Albert Visser, exploring some technical results in the field of non-standard models of arithmetic and their connections with satisfaction and truth. As it turns out, some PA systems to which the truth predicate is added will be conservative from the point of view of the theorems which can be proved in them, i.e. they are proof-conservative, but they will not be conservative in terms of the models that satisfy them, i.e. the extension in question is not model-conservative. This is significant, as the addition of a truth predicate is clearly interfering with the ‘substance’ of the relation of satisfiability of the theory in question with its models. So even if the addition of a truth predicate to a theory still guarantees proof-conservativeness (and in simple cases this does happen), it may still be adding ‘substance’ to the theory in terms of the models that will satisfy it.
A lot of the technical work presented at the conference still struck me as not having immediate philosophical significance (but of course, it might be just me missing something obvious). Nevertheless, what really surprised me was having the feeling that at least some of the philosophical work presented did not have immediate philosophical significance either. Ok, this will require some explanation, so here it goes.
The talks at the conference could be divided into three main groups: those adopting a formal perspective, those adopting a philosophical perspective, and those adopting a naturalistic perspective. What does the latter mean? Some of the speakers (Sheard, Hinzen, Collins) adopted a naturalistic perspective in that they focused on actual uses of the truth predicate in linguistic practices: why would we need a truth predicate? How do we actually use it? What do we gain by using a truth predicate? Obviously, answers to such questions will be at least partially based on empirical elements; they may rely on linguistic data or on data concerning cognition. W. Hinzen, for example, argued that the concept of truth must be approached from a grammatical point of view, rather than a semantic one, namely in terms of the grammatical constructions where the truth predicate occurs.
Now, the claim that the real target phenomenon of both philosophical and axiomatic theories of truth is the concept of truth underlying actual discursive practices, truth-in-use so to speak, is not particularly controversial; I suspect most people would agree with it. But a bit to my surprise, it would seem that philosophical approaches to truth are just as prone to becoming disconnected from their target-phenomenon as axiomatic approaches to truth. They may become equally baroque, albeit in different ways, if they lose sight of the actual phenomena they seek to explain, and that seems to be happening to a number of philosophical theories of truth -- it will come as no surprise to those who have read my previous posts that I endorse an empirically-informed approach in philosophy. (But as an aside, let me note that I’ve done a bit of work on truth myself (here is my most recent paper on the topic), which is absolutely not empirically-informed (it’s mostly logical and historical), so I would not want to claim that an empirically-informed approach is the only way to go about when it comes to truth.)
So what could constitute a fruitful naturalistic/philosophical approach to truth? It occurred to me that ‘truth’ (or the ‘is true’ predicate) is a fairly high-order concept (as also evidenced by its behavior as a second-order quantifier in axiomatic theories of truth), which as such might emerge in linguistic practices only given some background conditions. The analogy I have in mind is with words for numbers beyond very small numbers (i.e. up to three), which we now know not to exist in quite a few cultures/languages of the world (see the work of Pierre Pica and collaborators). My question to Hinzen after his talk was whether he had cross-linguistic data on the presence or absence of truth predicates in different languages; he replied that, to his knowledge, all human spoken languages possess a truth predicate, but he admitted not having data concerning ‘weird’ languages such as those not containing words for numbers (e.g. the famous Pirahã). My hypothesis is that there could well be perfectly functional human languages lacking a truth predicate (or anything resembling one), and it seems to me that this would be a fact of philosophical significance. It might at first sight seem to corroborate the deflationist position (since some linguistic and social practices would be doing fine without the truth predicate), but again the analogy with words for numbers suggests that things might not be so simple. We do know of communities of speakers who do not use words for numbers beyond very small numbers and are doing fine in their environment, but there is no doubt that having words for numbers represents a major cognitive boost to those who master this technology (Helen de Cruz and Johan de Smedt have a great paper on an extended cognition perspective on words and symbols for numbers). In a sense, again something analogous to what has been observed in axiomatic theories of truth may be going on: the truth predicate, even if metaphysically deflated and 'unsubstantial', seems to add considerable expressive and provability power to a theory. Even when the extension of a theory U with a truth predicate is still proof-conservative, in some cases (and these are technical results also discussed by Albert Visser) one observes the phenomenon known as ‘speed-up’, i.e. the proofs may get significantly shorter.
So from an ‘extended cognition’ perspective, an important philosophical/naturalistic question to be asked about truth seems to concern the potential cognitive and expressive boost provided by the mastery of a truth predicate, viewed as a higher-order concept. Data to be taken into account would be cross-linguistic studies on how the truth predicate behaves in different languages (including the possibility of it not existing in some languages) as well as developmental data on how young children develop the concept of truth (perhaps looking into the false-belief task materials, as once suggested by my colleague Dora Achourioti). (There might be some connection between these ideas and Brandom’s expressivist project.)
For now, these are inconsequential musings, but perhaps they could provide a starting point for future work? Time will tell, and truth will be told :)
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